Generator (category theory)

In mathematics, specifically category theory, a family of generators (or family of separators) of a category ${\displaystyle {\mathcal {C}}}$ is a collection ${\displaystyle {\mathcal {G}}\subseteq Ob({\mathcal {C}})}$ of objects in ${\displaystyle {\mathcal {C}}}$, such that for any two distinct morphisms ${\displaystyle f,g:X\to Y}$ in ${\displaystyle {\mathcal {C}}}$, that is with ${\displaystyle f\neq g}$, there is some ${\displaystyle G}$ in ${\displaystyle {\mathcal {G}}}$ and some morphism ${\displaystyle h:G\to X}$ such that ${\displaystyle f\circ h\neq g\circ h.}$ If the collection consists of a single object ${\displaystyle G}$, we say it is a generator (or separator).

Generators are central to the definition of Grothendieck categories.

The dual concept is called a cogenerator or coseparator.

Examples

• In the category of abelian groups, the group of integers ${\displaystyle \mathbf {Z} }$ is a generator: If f and g are different, then there is an element ${\displaystyle x\in X}$, such that ${\displaystyle f(x)\neq g(x)}$. Hence the map ${\displaystyle \mathbf {Z} \rightarrow X,}$ ${\displaystyle n\mapsto n\cdot x}$ suffices.
• Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator.
• In the category of sets, any set with at least two elements is a cogenerator.
• In the category of modules over a ring R, a generator in a finite direct sum with itself contains an isomorphic copy of R as a direct summand. Consequently, a generator module is faithful, i.e. has zero annihilator.

References

• Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2, p. 123, section V.7

External links

• separator at the nLab